A spokesperson for Fraport told City A.M. previous discussions, which took place during 2014, would recommence.
"The government [of Greece] has indicated its intent," he added.
Reuters reported that a Greek government official had confirmed the deal, thought to be worth €1.2bn (£845m).
The original talks took place last year, although were called into doubt when Prime Minister Alexis Tsipras' far left Syriza Party took power at the beginning of this year. However, in May it was reported that the government was keen to conclude the discussions "immediately".
This time round, Tsipras is under more pressure, after Greece signed up to a tough set of reforms in exchange for a bailout worth €86bn. The deal, which was voted through by the Greek parliament after a marathon all-night debate on Friday morning, includes tough privatisation measures.